代写代考 FM321: Risk Management and Zhu

Leture 2: Univariate Volatility Modelling – Part I
FM321: Risk Management and Zhu
4 October 2022
LSE Finance

LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 1 / 32

Main topics
Introduction to financial returns and risks
Volatility as the simplest risk measure
Volatility is useful for various contexts:
Investment decisions Portfolio construction Risk management Derivatives pricing
Other risk measures beyond volatility Implementing risk forecasts Backtesting and stress-testing
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 2 / 32

Volatility modelling
Univariate volatility modelling (one financial asset)
Moving average models ARCH
Model estimation Diagnostics
Alternative approaches
Multivariate volatility modelling
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 3 / 32

Unconditional vs. conditional variance
Recall two different concepts of variance for a series {rt } Unconditional variance
variance of rt given no information
σ2 =Var(rt)
Conditional variance
variance of rt given past information
σt2 = Var(rt|rt−1,rt−2,…)
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 4 / 32

Estimation of unconditional variance
Unconditional variance can be estimated by the sample variance of returns
σˆ 2 = T − 1
when rt is stationary (implying that the unconditional variance is a
constant).
For high-frequency returns (such as daily returns), r ̄ ≈ 0, so
σˆ 2 ≈ T − 1
( r t − r ̄ ) 2
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 5 / 32

Models of conditional variance
σt2 = Var(rt|rt−1,rt−2,…) ← need a model to estimate What features of data should the model capture?
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 6 / 32

Models of conditional variance
σt2 = Var(rt|rt−1,rt−2,…) ← need a model to estimate What features of data should the model capture?
Volatility clusters
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 6 / 32

Models of conditional variance
σt2 = Var(rt|rt−1,rt−2,…) ← need a model to estimate What features of data should the model capture?
Volatility clusters
An unexpected shock to returns is usually followed by a period of high conditional volatility
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 6 / 32

Models of conditional variance
σt2 = Var(rt|rt−1,rt−2,…) ← need a model to estimate What features of data should the model capture?
Volatility clusters
An unexpected shock to returns is usually followed by a period of high conditional volatility
High conditional volatility in a period is followed by high conditional volatility in the next few periods, but shocks eventually die out
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 6 / 32

Moving average model: a starting point
Let WE denote the estimation window.
The conditional variance is the average sum of squared returns over
the estimation window:
What’s good about the model?
1 WE σˆ2= Xr2
t WE t−i i=1
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 7 / 32

Moving average model: a starting point
Let WE denote the estimation window.
The conditional variance is the average sum of squared returns over
the estimation window:
1 WE σˆ2= Xr2
t WE t−i i=1
What’s good about the model?
Adaptive model that captures some of the desired properties (shocks to return lead to higher conditional variance, but eventually die out).
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 7 / 32

Moving average model: a starting point
Let WE denote the estimation window.
The conditional variance is the average sum of squared returns over
the estimation window:
1 WE σˆ2= Xr2
t WE t−i i=1
What’s good about the model?
Adaptive model that captures some of the desired properties (shocks to return lead to higher conditional variance, but eventually die out).
Easy to implement (no parameter to be estimated).
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 7 / 32

Moving average model: a starting point
What’s wrong with the model?
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 8 / 32

Moving average model: a starting point
What’s wrong with the model? Sensitive to the estimation window WE .
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 8 / 32

Moving average model: a starting point
What’s wrong with the model? Sensitive to the estimation window WE .
When WE is small, model specification leads to abrupt changes in conditional volatility estimates when there is a return shock without a true volatility shock.
When WE is large, conditional volatility estimate react too slowly to a true volatility shock and also die out too slowly.
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 8 / 32

Moving average model: a starting point
What’s wrong with the model? Sensitive to the estimation window WE .
When WE is small, model specification leads to abrupt changes in conditional volatility estimates when there is a return shock without a true volatility shock.
When WE is large, conditional volatility estimate react too slowly to a true volatility shock and also die out too slowly.
Equally weighted scheme cannot capture volatility clustering.
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 8 / 32

Moving average model: a starting point
What’s wrong with the model? Sensitive to the estimation window WE .
When WE is small, model specification leads to abrupt changes in conditional volatility estimates when there is a return shock without a true volatility shock.
When WE is large, conditional volatility estimate react too slowly to a true volatility shock and also die out too slowly.
Equally weighted scheme cannot capture volatility clustering.
Exponential decay was introduced to remove some of the drawbacks of moving average models
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 8 / 32

Exponentially-weighted moving average model (EWMA)
Model specification:
σ2 =(1−λ)r2 +λσ2
λ: decay factor satisfying 0 < λ < 1 Higher λ implies greater persistence of the impact of shocks. LSE FM321 Leture 2: Univariate Volatility Modelling - Part I 9 / 32 The idea of EWMA Allows for greater influence of more recent observations on volatility estimates than more distant ones. Conditional variance is a weighted sum of past squared returns, σ2 =w r2 +w r2 +···+w t 1 t−1 2 t−2 and the weights are exponentially declining: w2 = λw1 w3 = λ2w1 w4 = λ3w1 ... wk = λk−1w1 Leture 2: Univariate Volatility Modelling - Part I The idea of EWMA w1 is an appropriate normalizing constant chosen so that the weights sum up to 1. 1=Xwi =w1 +λw1 +λ2w1 +...λWE−1w1 =(1+λ+λ2 +···+λWE−1)w1 1−λWE = 1−λ w1 w1= 1−λ σ2 = 1−λ (r2 +λr2 +···+λWE−1r2 ) t 1−λWE t−1 t−2 t−WE 1−λ WE Leture 2: Univariate Volatility Modelling - Part I λ(1 − λWE ) EWMA: derivation How to get the EWMA equation: σ2 =(1−λ)r2 +λσ2 LSE FM321 Leture 2: Univariate Volatility Modelling - Part I 12 / 32 EWMA: derivation How to get the EWMA equation: σ2 =(1−λ)r2 +λσ2 When WE is large, λi becomes negligible for all i ≥ WE . Hence, approximate the model by setting WE → ∞ ∞ σ2 =(1−λ)Xλi−1r2 =(1−λ)r2 +P∞ λi−1r2 = (1 − λ)r 2 = (1 − λ)r 2 = (1 − λ)r2 t−1 + λ (1 − λ) X λi −1 r 2 t−i−1 t−1 i=2 t−1 + (1 − λ) X λi −1 r 2 Leture 2: Univariate Volatility Modelling - Part I EWMA: what value does λ take? EWMA was implemented in large scale by JP Morgan in the late 1980s and early 1990s, and made available broadly under the brand name RiskMetrics (around the same time as the concept of Value-at-Risk) JP Morgan set for daily data: λ = 0.94 When we use the term EWMA we are assuming that value. LSE FM321 Leture 2: Univariate Volatility Modelling - Part I 13 / 32 Unconditional EWMA variance is not defined! Suppose a return series {rt} satisfies: σ2 =(1−λ)r2 +λσ2 LSE FM321 Leture 2: Univariate Volatility Modelling - Part I 14 / 32 Unconditional EWMA variance is not defined! Suppose a return series {rt} satisfies: σ2 =(1−λ)r2 +λσ2 For a scaled version of the return, rt′ = crt (c > 0):
(σt′)2 = Var[(r′)2t |rt′−1,…] = Var[crt2|crt−1, . . . ]
= c2Var(rt2|rt−1, . . . ) = c 2 σ t2
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 14 / 32

Unconditional EWMA variance is not defined!
Suppose a return series {rt} satisfies:
σ2 =(1−λ)r2 +λσ2
For a scaled version of the return, rt′ = crt (c > 0):
(σt′)2 = Var[(r′)2t |rt′−1,…] = Var[crt2|crt−1, . . . ]
= c2Var(rt2|rt−1, . . . ) = c 2 σ t2
Hence, rt′ also satisfies the same EWMA equation above.
(σ′)2 =c2σ2 =c2[(1−λ)r2 +λσ2 ]=(1−λ)(r′ )2 +λ(σ′ )2
t t t−1 t−1 t−1 t−1
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 14 / 32

Unconditional EWMA variance is not defined!
Suppose a return series {rt} satisfies:
σ2 =(1−λ)r2 +λσ2
For a scaled version of the return, rt′ = crt (c > 0):
unconditional variance of the model.
(σt′)2 = Var[(r′)2t |rt′−1,…] = Var[crt2|crt−1, . . . ]
= c2Var(rt2|rt−1, . . . ) = c 2 σ t2
Hence, rt′ also satisfies the same EWMA equation above.
(σ′)2 =c2σ2 =c2[(1−λ)r2 +λσ2 ]=(1−λ)(r′ )2 +λ(σ′ )2
t t t−1 t−1 t−1 t−1 Knowledge of model parameters is insufficient to pin down the
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 14 / 32

Two questions from last time
Why do we care whether we can or cannot characterize the unconditional variance of EWMA, or any volatility model?
What exactly is in the specification of EWMA that makes the unconditional variance non-existent?
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 15 / 32

A preview of the answers
Unconditional variance resembles the level that we would expect the conditional variance to stay at if there were no shocks to the system.
Once a shock hits, conditional variance will rise, and eventually mean-revert to the original level.
Non-existence of the unconditional variance suggests that future conditional variances will instead drift away like a random walk.
Key problem: the sum of coefficients is 1, and there is no constant term.
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 16 / 32

EWMA – effect of a shock
What happens to Et[σ2 ] when there is a shock to σt2? t+k
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 17 / 32

Standardized residual
Write log return rt as
where zt is called the standardized residual satisfying Et−1[zt] = 0
and Vart−1[zt ] = 1.
zt represents the component of return that is independent of all
information up to time t − 1.
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 18 / 32

Standardized residual
Write log return rt as
where zt is called the standardized residual satisfying Et−1[zt] = 0
and Vart−1[zt ] = 1.
zt represents the component of return that is independent of all
information up to time t − 1.
Verify that σt2 is indeed the conditional variance of rt :
Vart−1(rt) = Et−1(rt2) = Et−1(σt2zt2) = σt2Et−1(zt2) = σt2
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 18 / 32

Properties of the standardized residual
For any future date t + k (k > 0), we have E[r2 ]=E[σ2 z2 ]
= E [E [σ2 z2 ]]
t t+k−1 t+k t+k = E [σ2 E [z2
t t+k t+k−1 t+k =E[σ2 ]
Allow us to focus on modeling σt2, leaving zt as a standardized white noise that does not require modeling (other than assigning a distribution, usually the standard normal distribution).
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 19 / 32

EWMA – effect of a shock
Shocks to volatility in EWMA are not mean-reverting, so a jump in conditional volatility is not expected to come down. Reminds us of the random walk model (that you might have seen elsewhere), which does not have a stationary mean and has an infinite variance.
E [σ2 ]=E [σ2 ]=···=E [σ2 ]=σ2
t t+k t t+k−1
To see this, start from the EWMA equation:
]=(1−λ)E [r2
]+λE [σ2 ] t t+k−1
σ2 = (1 − λ)r2
+ λσ2 , t+k−1
=(1−λ)E [σ2
= E [σ2 ] t t+k−1
Leture 2: Univariate Volatility Modelling – Part I

EWMA – effect of a shock
Shocks to volatility in EWMA are not mean-reverting, so a jump in conditional volatility is not expected to come down. Reminds us of the random walk model (that you might have seen elsewhere), which does not have a stationary mean and has an infinite variance.
E [σ2 ]=E [σ2 ]=···=E [σ2 ]=σ2
t t+k t t+k−1
To see this, start from the EWMA equation:
]=(1−λ)E [r2
]+λE [σ2 ] t t+k−1
σ2 = (1 − λ)r2
+ λσ2 , t+k−1
=(1−λ)E [σ2
= E [σ2 ] t t+k−1
Key problem: the sum of coefficients is 1, and there is no constant term.
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 20 / 32

Autoregressive conditional heteroskedasticity (ARCH)
ARCH models were proposed by Engle (1982)
Bollerslev (1986) generalized ARCH to GARCH (Generalized ARCH) models.
Serve as the basis for most univariate volatility models used in practice
Model conditional volatility as a function of:
Past squared returns (in ARCH);
Past conditional volatility (in GARCH);
Possibly additional factors (in more complex specifications)
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 21 / 32

ARCH – model specification
The defining equation of an ARCH(1) process is
More generally, ARCH(N) allows for an arbitrary number N of lags:
σ2 =ω+Xα r2
It is necessary to assume that ω > 0, αn ≥ 0, and at least one of the αn is strictly positive.
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 22 / 32

ARCH – comments
ARCH models effectively incorporate the effects of past shocks on current volatility estimates.
They improve on moving average models in that, under appropriate conditions, they do not suffer the drawbacks related to unconditional variance.
LSE FM321 Leture 2: Univariate Volatility Modelling – Part I 23 / 32

ARCH(1) – unconditional variance
Assume that the ARCH process has a finite unconditional variance, and denote it by σ2.
Recall that E[rt2] = E[σt2] = σ2 for every t. Therefore, for an ARCH(1) process:
E[σ2 ]=ω+αE[r2] t+1 t
⇒σ2 =ω+ασ2 ⇒σ2= ω
In order for the unconditional variance of the process to exist, we
need α < 1. LSE FM321 Leture 2: Univariate Volatility Modelling - Part I 24 / 32 ARCH(N) - unconditional variance A similar argument can be used to establish that, for an ARCH(N) 2ω process we have: σ = 1−PNn=1 αn . In order for the unconditional variance of the process to exist, we need PNn=1 αn < 1 Unlike with EWMA, knowledge of the model parameters is sufficient to pin down the unconditional variance of the model LSE FM321 Leture 2: Univariate Volatility Modelling - Part I 25 / 32 ARCH(1) - conditional variance Therefore, for an ARCH(1) process: σ2 = ω + αr2 ] = ω + αE [r2 ] t t+k−1 ] ]=σ2(1−α)+αE [σ2 ] ] = ω + αE [σ2 t t+k−1 E[σ2 ]−σ2 =αE[σ2 ]−σ2 t t+k t t+k−1 Leture 2: Univariate Volatility Modelling - Part I ARCH(1) - conditional variance Therefore, for an ARCH(1) process: ]−σ2 =αk−1E[σ2 ]−σ2=αk−1(σ2 −σ2) t+1 ] = ω + αE [r2 ] t t+k−1 ] ]=σ2(1−α)+αE [σ2 ] σ2 = ω + αr2 ] = ω + αE [σ2 t t+k−1 E[σ2 ]−σ2 =αE[σ2 ]−σ2 Leture 2: Univariate Volatility Modelling - Part I ARCH(1) - conditional variance Therefore, for an ARCH(1) process: ]−σ2 =αk−1E[σ2 ]−σ2=αk−1(σ2 −σ2) t+1 ] = ω + αE [r2 ] t t+k−1 ] ]=σ2(1−α)+αE [σ2 ] σ2 = ω + αr2 Conditional variance forecasts are mean reverting. ] = ω + αE [σ2 t t+k−1 E[σ2 ]−σ2 =αE[σ2 ]−σ2 LSE FM321 Leture 2: Univariate Volatility Modelling - Part I ARCH(1) - tail behavior What can we say about the tail behavior of an ARCH(1) process? Assume that zt is normally distributed conditional on information up to time t − 1, so that Et−1[zt4] = 3 Note that E[rt4] = E[Et−1[rt4]] = E[Et−1[σt4zt4]] = E[σt4Et−1[zt4]] = 3 E [ σ t4 ] Leture 2: Univariate Volatility Modelling - Part I ARCH(1) - tail behavior σ4 = ω2 + 2αωr2 + α2r4 t−1 E[σ4]=ω2 +2αω ω +3α2E[σ4 ] 4 2h 1+α i (1−3α2)E[σ4]=ω21+ 2α t 1−α E[σt ] = ω (1−α)(1−3α2) We can compute the unconditional kurtosis as E [ r t4 ] 2 h 1 + α i ( 1 − α ) 2 E[rt2]2 = 3ω (1−α)(1−3α2) × ω2 = 3 (1 − α2) (1 − 3α2) LSE FM321 Leture 2: Univariate Volatility Modelling - Part I ARCH(1) - tail behavior In order for the kurtosis to exist, we need 3α2 < 1, or α < 0.577 So long as this condition is satisfied, the unconditional kurtosis is greater than 3 The ARCH(1) process has unconditionally heavy tails even though we assumed that the standardized residuals are conditionally normally distributed LSE FM321 Leture 2: Univariate Volatility Modelling - Part I 29 / 32 ARCH vs. EWMA LSE FM321 Leture 2: Univariate Volatility Modelling - Part I 30 / 32 ARCH vs. EWMA Both processes allow for temporary shocks to returns to have an effect on conditional volatility. LSE FM321 Leture 2: Univariate Volatility Modelling - Part I 30 / 32 ARCH vs. EWMA Both processes allow for temporary shocks to returns to have an effect on conditional volatility. However, ARCH can replicate some usual features of financial time series which EWMA cannot: Unconditional variance is finite. Shocks to volatility eventually die out. Tails are heavy even if standardized residuals are normal (if parameter values are such that fourth moment exists). LSE FM321 Leture 2: Univariate Volatility Modelling - Part I 30 / 32 ARCH(1): very unstable estimates LSE FM321 Leture 2: Univariate Volatility Modelling - Part I 31 / 32 ARCH - drawbacks In spite of all of the improvements over EWMA, ARCH models are not quite satisfactory (almost nobody uses it). Volatility estimates tend to be very unstable, which creates practical problems. This problem can be solved if we include a large number of lags, but that leads to estimation difficulties. In addition, conditions on parameters for existence of fourth moments tend to be too restrictive. LSE FM321 Leture 2: Univariate Volatility Modelling - Part I 32 / 32 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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